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Photonic crystal wave properties. Photonic Crystal




According to the nature of the change in the refractive index, photonic crystals can be divided into three main classes:

1. One-dimensional, in which the refractive index periodically changes in one spatial direction, as shown in Figure 2. In this figure, the symbol L denotes the period of change in the refractive index, and are the refractive indices of two materials (but in general, any number of materials can be present). Such photonic crystals consist of layers of different materials parallel to each other with different refractive indices and can exhibit their properties in one spatial direction perpendicular to the layers.

Figure 1 - Schematic representation of a one-dimensional photonic crystal

2. Two-dimensional, in which the refractive index periodically changes in two spatial directions as shown in Figure 2. In this figure, a photonic crystal is created by rectangular regions with a refractive index that are in a medium with a refractive index. In this case, the regions with the refractive index are ordered in a two-dimensional cubic lattice. Such photonic crystals can exhibit their properties in two spatial directions, and the shape of the regions with the refractive index is not limited to rectangles, as in the figure, but can be any (circles, ellipses, arbitrary, etc.). The crystal lattice in which these regions are ordered can also be different, and not just cubic, as in the figure.

Figure - 2 Schematic representation of a two-dimensional photonic crystal

3. Three-dimensional, in which the refractive index periodically changes in three spatial directions. Such photonic crystals can exhibit their properties in three spatial directions, and they can be represented as an array of volumetric regions (spheres, cubes, etc.) ordered in a three-dimensional crystal lattice.

Like electrical media, depending on the width of the forbidden and allowed zones, photonic crystals can be divided into conductors - capable of conducting light over long distances with low losses, dielectrics - almost perfect mirrors, semiconductors - substances capable, for example, of selectively reflecting photons of a certain wavelength and superconductors, in which, thanks to collective phenomena, photons are able to propagate over practically unlimited distances.

There are also resonant and non-resonant photonic crystals. Resonant photonic crystals differ from non-resonant ones in that they use materials whose permittivity (or refractive index) as a function of frequency has a pole at some resonant frequency.

Any inhomogeneity in a photonic crystal is called a photonic crystal defect. In such regions, the electromagnetic field is often concentrated, which is used in microresonators and waveguides based on photonic crystals.

Like electrical media, depending on the width of the forbidden and allowed zones, photonic crystals can be divided into conductors - capable of conducting light over long distances with low losses, dielectrics - almost perfect mirrors, semiconductors - substances capable, for example, of selectively reflecting photons of a certain wavelength and superconductors, in which, thanks to collective phenomena, photons are able to propagate over practically unlimited distances. There are also resonant and non-resonant photonic crystals. Resonant photonic crystals differ from non-resonant ones in that they use materials whose permittivity (or refractive index) as a function of frequency has a pole at some resonant frequency.

Any inhomogeneity in a photonic crystal is called a photonic crystal defect. In such regions, the electromagnetic field is often concentrated, which is used in microresonators and waveguides based on photonic crystals. There are a number of analogies in describing the propagation of electromagnetic waves in photonic crystals and the electronic properties of crystals. Let's take a look at some of them.

1. The state of an electron inside a crystal (the law of motion) is given by the solution of the Schrldinger equation, the propagation of light in a photonic crystal obeys the wave equation, which is a consequence of the Maxwell equations:

  • 2. The state of an electron is described by a scalar wave function w(r, t), the state of an electromagnetic wave is described by vector fields - the strength of the magnetic or electrical component, H (r, t) or E (r, t).
  • 3. The electron wave function w(r,t) can be expanded into a series of eigenstates wE(r), each of which corresponds to its own energy E. The electromagnetic field strength H(r,t) can be represented by a superposition of monochromatic components (modes) electromagnetic field Hsh(r), each of which corresponds to its own value - the frequency of the mode w:

4. Atomic potential U(r) and permittivity е(r), appearing in the Schrldinger and Maxwell equations, are periodic functions with periods equal to any vectors R of the crystal lattice and photonic crystal, respectively:

U(r) = U(r + R), (3)

5. For the wave function of the electron and the strength of the electromagnetic field, the Bloch theorem is satisfied with periodic functions u k and u k .

  • 6. Possible values ​​of wave vectors k fill the Brillouin zone of a crystal lattice or a unit cell of a photonic crystal, which is specified in the space of inverse vectors.
  • 7. The electron energy E, which is the eigenvalue of the Schrldinger equation, and the eigenvalue of the wave equation (consequences of the Maxwell equations) - the mode frequency u - are related to the values ​​of the wave vectors k of the Bloch functions (4) by the dispersion law E(k) and u(k).
  • 8. An impurity atom that breaks the translational symmetry of the atomic potential is a crystal defect and can create an impurity electronic state localized in the vicinity of the defect. Changes in the permittivity in a certain region of the photonic crystal break the translational symmetry e(r) and lead to the appearance of an allowed mode inside the photonic band gap localized in its spatial vicinity.

Photonic crystals (PCs) are structures characterized by a periodic change in the permittivity in space. The optical properties of PCs are very different from the optical properties of continuous media. The propagation of radiation inside a photonic crystal, due to the periodicity of the medium, becomes similar to the movement of an electron inside an ordinary crystal under the action of a periodic potential. As a result, electromagnetic waves in photonic crystals have a band spectrum and a coordinate dependence similar to the Bloch waves of electrons in ordinary crystals. Under certain conditions, gaps form in the band structure of a PC, similarly to forbidden electronic bands in natural crystals. Depending on the specific properties (the material of the elements, their size, and the grating period), the PC spectrum can form both completely frequency-forbidden zones, for which radiation propagation is impossible regardless of its polarization and direction, and partially forbidden (stop-zones), in which can spread only in selected directions.

Photonic crystals are of interest both from a fundamental point of view and for numerous applications. On the basis of photonic crystals, optical filters, waveguides (in particular, in fiber-optic communication lines), devices that allow controlling thermal radiation are created and developed, laser designs with a lower pump threshold have been proposed based on photonic crystals.

In addition to changing the reflection, transmission and absorption spectra, metal-dielectric photonic crystals have a specific density of photonic states. The changed density of states can significantly affect the lifetime of the excited state of an atom or molecule placed inside a photonic crystal and, consequently, change the nature of the luminescence. For example, if the transition frequency in an indicator molecule located in a photonic crystal falls into the band gap, then luminescence at this frequency will be suppressed.

FCs are divided into three types: one-dimensional, two-dimensional and three-dimensional.

One-, two- and three-dimensional photonic crystals. Different colors correspond to materials with different dielectric constants.

One-dimensional are PCs with alternating layers made of different materials.


Electron image of a one-dimensional PC used in a laser as a Bragg multilayer mirror.

Two-dimensional FKs can have more diverse geometries. These include, for example, arrays of cylinders of infinite length (their transverse size is much smaller than the longitudinal one) or periodic systems of cylindrical holes.


Electronic images, two-dimensional forward and reverse FK with a triangular lattice.

The structures of three-dimensional PCs are very diverse. The most common in this category are artificial opals - ordered systems of spherical diffusers. There are two main types of opals: straight and reverse (inverse) opals. The transition from direct opal to reverse opal is carried out by replacing all spherical elements with cavities (usually air), while the space between these cavities is filled with some material.

Below is the surface of a PC, which is a straight opal with a cubic lattice based on self-organized spherical polystyrene microparticles.


The inner surface of a PC with a cubic lattice based on self-organized spherical polystyrene microparticles.

The next structure is an inverse opal synthesized as a result of a multi-stage chemical process: self-assembly of polymer spherical particles, impregnation of voids in the resulting material with a substance, and removal of the polymer matrix by chemical etching.


The surface of a quartz inverse opal. The photograph was obtained using scanning electron microscopy.

Another type of three-dimensional FCs are structures of the "woodpile" type (logpiles), formed by rectangular parallelepipeds crossed, as a rule, at right angles.


Electronic photo of PC from metal parallelepipeds.

Production Methods

The use of FCs in practice is significantly limited by the lack of universal and simple methods for their manufacture. In our time, several approaches to the creation of a FC have been implemented. Two main approaches are described below.

The first of these is the so-called self-organization or self-assembly method. When self-assembling a photonic crystal, colloidal particles are used (the most common are monodisperse silicon or polystyrene particles), which are in the liquid and, as the liquid evaporates, are deposited in the volume. As they "deposit" on each other, they form a three-dimensional PC and are ordered, depending on the conditions, into a cubic face-centered or hexagonal crystal lattice. This method is quite slow, the formation of FC may take several weeks. Also, its disadvantages include a poorly controlled percentage of the appearance of defects in the deposition process.

One of the varieties of the self-assembly method is the so-called honeycomb method. This method involves filtering the liquid in which the particles are located through small pores, and allows the formation of FC at a rate determined by the rate of flow of the liquid through these pores. Compared with the conventional deposition method, this method is much faster, however, the percentage of defects in its use is also higher.

The advantages of the described methods include the fact that they allow the formation of PC samples of large sizes (with an area of ​​up to several square centimeters).

The second most popular method for the manufacture of FC is the etching method. Various etching methods are generally used to fabricate 2D PCs. These methods are based on the use of a photoresist mask (which defines, for example, an array of hemispheres) formed on the surface of a dielectric or metal and defining the geometry of the etched region. This mask can be obtained using the standard photolithography method, followed directly by chemical etching of the sample surface with photoresist. In this case, respectively, in the areas where the photoresist is located, the surface of the photoresist is etched, and in the areas without a photoresist, the dielectric or metal is etched. The process continues until the desired etch depth is reached, after which the photoresist is washed off.

The disadvantage of this method is the use of the photolithography process, the best spatial resolution of which is determined by the Rayleigh criterion. Therefore, this method is suitable for creating a PC with a band gap, which, as a rule, lies in the near infrared region of the spectrum. Most often, a combination of photolithography with electron beam lithography is used to achieve the desired resolution. This method is an expensive but highly accurate method for fabricating quasi-two-dimensional PCs. In this method, a photoresist that changes its properties under the action of an electron beam is irradiated at specific locations to form a spatial mask. After irradiation, part of the photoresist is washed off, and the remaining part is used as an etching mask in the subsequent technological cycle. The maximum resolution of this method is about 10 nm.

Parallels between electrodynamics and quantum mechanics

Any solution of Maxwell's equations , in the case of linear media and in the absence of free charges and current sources, can be represented as a superposition of functions harmonic in time with complex amplitudes depending on frequency: , where is either , or .

Since the fields are real, then , and can be written as a superposition of functions harmonic in time with a positive frequency: ,

Consideration of harmonic functions allows us to pass to the frequency form of Maxwell's equations, which does not contain time derivatives: ,

where the time dependence of the fields involved in these equations is represented as , . We assume that the media are isotropic and that the magnetic permeability is .

Explicitly expressing the field, taking the curl from both sides of the equations, and substituting the second equation into the first, we get:

where is the speed of light in vacuum.

In other words, we got an eigenvalue problem:

for the operator

where the dependence is determined by the structure under consideration.

The eigenfunctions (modes) of the resulting operator must satisfy the condition

Located as

In this case, the condition is met automatically, since the divergence of the rotor is always zero.

The operator is linear, which means that any linear combination of solutions to the eigenvalue problem with the same frequency will also be a solution. It can be shown that in the case this operator is Hermitian, i.e., for any vector functions

where the dot product is defined as

Since the operator is Hermitian it follows that its eigenvalues ​​are real. It can also be shown that at 0" align="absmiddle">, the eigenvalues ​​are non-negative, and hence the frequencies are real.

The scalar product of the eigenfunctions corresponding to different frequencies is always zero. In the case of equal frequencies, this is not necessarily the case, but it is always possible to work only with mutually orthogonal linear combinations of such eigenfunctions. Moreover, it is always possible to form a basis from mutually orthogonal eigenfunctions of the Hermitian operator .

If, on the contrary, we express the field in terms of , we get a generalized eigenvalue problem:

in which operators are already present on both sides of the equation (in this case, after division by the operator on the left side of the equation, it becomes non-Hermitian). In some cases, this formulation is more convenient.

Note that when the equation is replaced by eigenvalues, the frequency will correspond to the new solution. This fact is called scalability and is of great practical importance. The production of photonic crystals with characteristic dimensions on the order of a micron is technically difficult. However, for testing purposes, it is possible to make a model of a photonic crystal with a period and an element size of the order of a centimeter that would operate in centimeter mode (in this case, materials should be used that would have approximately the same permittivity in the centimeter frequency range as the simulated materials).

Let us draw an analogy of the theory described above with quantum mechanics. In quantum mechanics, a scalar wave function is considered that takes complex values. In electrodynamics, it is vector, and the complex dependence is introduced only for convenience. A consequence of this fact, in particular, is that the band structures for photons in a photonic crystal will be different for waves with different polarizations, in contrast to the band structures for electrons.

Both in quantum mechanics and in electrodynamics, the problem is solved for the eigenvalues ​​of the Hermitian operator. In quantum mechanics, Hermitian operators correspond to observables.

And finally, in quantum mechanics, if the operator is represented as a sum , the solution of the eigenvalue equation can be written as , that is, the problem is divided into three one-dimensional ones. In electrodynamics, this is impossible, since the operator "links" all three coordinates, even if they are separated in. For this reason, only a very limited number of problems in electrodynamics have analytical solutions. In particular, exact analytical solutions for the band spectrum of a PC are found mainly for one-dimensional PCs. That is why numerical simulation plays an important role in calculating the properties of photonic crystals.

Band structure

The photonic crystal is characterized by the periodicity of the function:

An arbitrary translation vector represented as

where are primitive translation vectors and are integers.

By Bloch's theorem, the eigenfunctions of an operator can be chosen in such a way that they have the form of a plane wave multiplied by a function that has the same periodicity as the FK:

where is a periodic function. In this case, the values ​​can be selected in such a way that they belong to the first Brillouin zone.

Substituting this expression into the formulated eigenvalue problem, we obtain an eigenvalue equation

Eigenfunctions must be periodic and satisfy the condition .

It can be shown that each value of the vector corresponds to an infinite set of modes with a discrete set of frequencies , which we will number in ascending order with the index . Since the operator depends continuously on , the frequency at a fixed index on also depends continuously. The set of continuous functions constitutes the band structure of the FK. The study of the band structure of a photonic crystal makes it possible to obtain information about its optical properties. The presence of any additional symmetry in the FK allows us to confine ourselves to a certain subdomain of the Brillouin zone, which is called irreducible. The solutions for , which belongs to this irreducible zone, reproduce the solutions for the entire Brillouin zone.


Left: A 2D photonic crystal made up of cylinders packed into a square lattice. Right: The first Brillouin zone corresponding to a square lattice. The blue triangle corresponds to the irreducible Brillouin zone. G, M and X- points of high symmetry for a square lattice.

Frequency intervals that do not correspond to any modes for any real value of the wave vector are called band gaps. The width of such zones increases with an increase in the contrast of the permittivity in a PC (the ratio of the permittivities of the constituent elements of a photonic crystal). If radiation with a frequency lying inside the forbidden band is generated inside such a photonic crystal, it cannot propagate in it (it corresponds to the complex value of the wave vector). The amplitude of such a wave will decay exponentially inside the crystal (evanescent wave). One of the properties of a photonic crystal is based on this: the possibility of controlling spontaneous emission (in particular, its suppression). If such radiation is incident on the PC from outside, then it is completely reflected from the photonic crystal. This effect is the basis for the use of PC for reflective filters, as well as for resonators and waveguides with highly reflective walls.

As a rule, low-frequency modes are concentrated mainly in layers with a large dielectric constant, while high-frequency modes are mostly concentrated in layers with a lower dielectric constant. Therefore, the first zone is often called the dielectric zone, and the one following it is called the air zone.


Band structure of a one-dimensional PC corresponding to wave propagation perpendicular to the layers. In all three cases, each layer has a thickness of 0.5 a, where a- FC period. Left: Each layer has the same permittivity ε = 13. Center: The permittivity of alternating layers has the values ε = 12 and ε = 13. Right: ε = 1 and ε = 13.

In the case of a PC with dimensions less than three, there are no complete band gaps for all directions, which is a consequence of the presence of one or two directions along which the PC is homogeneous. Intuitively, this can be explained by the fact that the wave does not experience multiple reflections along these directions, which is required for the formation of band gaps.

Despite this, it is possible to create one-dimensional PCs that would reflect waves incident on the PC at any angle.


The band structure of a one-dimensional PC with a period a, in which the thicknesses of alternating layers are 0.2 a and 0.8 a, and their permittivity - ε = 13 and ε = 1, respectively. The left part of the figure corresponds to the direction of wave propagation perpendicular to the layers (0, 0, k z), and the right one - in the direction along the layers (0, k y , 0). The band gap exists only for the direction perpendicular to the layers. Note that when k y > 0, the degeneracy is removed for two different polarizations.

The band structure of a PC with an opal geometry is shown below. It can be seen that this PC has a total band gap at a wavelength of about 1.5 µm and one stop band, with a reflection maximum at a wavelength of 2.5 µm. By varying the etching time of the silicon matrix at one of the stages of inverse opal fabrication and thus by varying the diameter of the spheres, it is possible to localize the band gap in a certain wavelength range. The authors note that a structure with similar characteristics can be used in telecommunication technologies. Radiation at the band gap frequency can be localized inside the volume of the PC, and when the necessary channel is provided, it can propagate virtually without loss. Such a channel can be formed, for example, by removing photonic crystal elements along a certain line. When the channel is bent, the electromagnetic wave will also change direction, repeating the shape of the channel. Thus, such a PC is supposed to be used as a transmission unit between an emitting device and an optical microchip that processes the signal.


Comparison of the reflectance spectrum in the GL direction, measured experimentally, and the band structure calculated by the plane wave expansion method for an inverse silicon (Si) opal with a face-centered cubic lattice (the inset shows the first Brillouin zone). The volume fraction of silicon is 22%. Grating period 1.23 µm

In the case of one-dimensional PCs, even the smallest permittivity contrast is sufficient to form a band gap. It would seem that for three-dimensional dielectric PCs, a similar conclusion can be drawn: to assume the presence of a complete bandgap at any small contrast of dielectric permittivity in the case if, at the boundary of the Brillouin zone, the vector has the same moduli in all directions (which corresponds to the spherical Brillouin zone). However, three-dimensional crystals with a spherical Brillouin zone do not exist in nature. As a rule, it has a rather complex polygonal shape. Thus, it turns out that band gaps in different directions exist at different frequencies. Only if the dielectric contrast is large enough can the stop bands in different directions overlap and form a complete band gap in all directions. Closest to spherical (and thus most independent of the direction of the Bloch vector ) is the first Brillouin zone of the face-centered cubic (fcc) and diamond lattices, making 3D PCs with this structure most suitable for forming a total band gap in the spectrum. At the same time, for the appearance of total band gaps in the spectra of such PCs, a large contrast in the dielectric constant is required. If we denote the relative slit width as , then to achieve the values ​​of 5\%" align="absmiddle">, a contrast is required for the diamond and fcc gratings, respectively. , bearing in mind that all PCs obtained in experiments are not ideal, and defects in the structure can significantly reduce the band gap.


The first Brillouin zone of a cubic face-centered lattice and points of high symmetry.

In conclusion, we note once again the similarity of the optical properties of PCs with the properties of electrons in quantum mechanics when considering the band structure of a solid. However, there is a significant difference between photons and electrons: electrons have a strong interaction with each other. Therefore, “electronic” problems, as a rule, require taking into account many-electron effects, which greatly increase the dimension of the problem, which often forces the use of insufficiently accurate approximations, while in a PC consisting of elements with a negligible nonlinear optical response, this difficulty is absent.

A promising area of ​​modern optics is the control of radiation with the help of photonic crystals. In particular, log-piles PCs were studied at the Sandia Laboratory in order to achieve high selectivity of the emission of metal photonic crystals in the near infrared range, simultaneously with strong suppression of radiation in the mid-IR range (<20мкм). В этих работах было показано, что для таких ФК излучение в среднем ИК диапазоне сильно подавлено из-за наличия в спектре ФК полной фотонной щели. Однако качество полной фотонной щели падает с ростом температуры из-за увеличения поглощения в вольфраме, что приводит к низкой селективности излучения при высоких температурах.

According to Kirchhoff's law for radiation in thermal equilibrium, the emissivity of a gray body (or surface) is proportional to its absorptivity. Therefore, in order to obtain information on the emissivity of metallic PCs, one can study their absorption spectra. To achieve high selectivity of the emitting structure in the visible range (nm) containing PC, it is necessary to choose such conditions under which the absorption in the visible range is large, and in the IR is suppressed.

In our works, http, we analyzed in detail the change in the absorption spectrum of a photonic crystal with elements of tungsten and with the geometry of opal with a change in all its geometric parameters: lattice period, size of tungsten elements, and the number of layers in a PC sample. An analysis was also made of the influence on the absorption spectrum of defects in a PC that arise during its manufacture.

I cannot claim to be impartial in judging colors. I rejoice in sparkling hues and sincerely regret the skimpy browns. (Sir Winston Churchill).

Origin of photonic crystals

Looking at the wings of a butterfly or the mother-of-pearl coating of shells (Figure 1), one wonders how Nature - even for many hundreds of thousands or millions of years - could create such amazing biostructures. However, not only in the bioworld there are similar structures with iridescent color, which are an example of the almost limitless creative possibilities of Nature. For example, the semi-precious stone opal has fascinated people since ancient times with its brilliance (Figure 2).

Today, every ninth grader knows that not only the processes of absorption and reflection of light lead to what we call the coloring of the world, but also the processes of diffraction and interference. Diffraction gratings that we can find in nature are structures with a periodically changing dielectric constant, while their period is commensurate with a long wavelength of light (Figure 3). These can be 1D lattices, as in the mother-of-pearl coating of mollusk shells such as galiotis, 2D lattices, like barnacles of sea mice, polychaete worms, and 3D lattices, which give iridescent blue coloring to butterflies from Peru, as well as opal.

In this case, Nature, as undoubtedly the most experienced materials chemist, is pushing us to the following solution: three-dimensional optical diffraction gratings can be synthesized by creating dielectric gratings that are geometrically complementary to each other, i.e. one is inverse to the other. And since Jean-Marie Lehn said the famous phrase: “If something exists, then it can be synthesized,” we simply have to put this conclusion into practice.

Photonic Semiconductors and the Photonic Gap

So, in a simple formulation, a photonic crystal is a material whose structure is characterized by a periodic change in the refractive index in spatial directions, which leads to the formation of a photonic band gap. Usually, in order to understand the meaning of the terms "photonic crystal" and "photonic bandgap", such a material is considered as an optical analogy to semiconductors. The solution of Maxwell's equations for the propagation of light in a dielectric grating shows that, due to Bragg diffraction, the distribution of photons over frequencies ω(k) depending on the wave vector k (2π/λ) will have discontinuity regions. This statement is graphically presented in Figure 4, which shows an analogy between the propagation of an electron in a 1D crystal lattice and a photon in a 1D photonic lattice. The continuous density of states of both a free electron and a photon in vacuum undergo a break inside, respectively, the crystal and photon lattices in the so-called "stop zones" at the value of the wave vector k (i.e. momentum), which corresponds to a standing wave. This is the condition for the Bragg diffraction of an electron and a photon.

The photonic band gap is a frequency range ω(k) in the reciprocal space of wave vectors k, where the propagation of light of a certain frequency (or wavelength) is prohibited in the photonic crystal in all directions, while the light incident on the photonic crystal is completely reflected from it. If the light "arises" inside the photonic crystal, then it will be "frozen" into it. The zone itself may be incomplete, the so-called stop zone. Figure 5 shows 1D, 2D and 3D photonic crystals in real space and the photon density of states in reciprocal space.

The photonic band gap of a three-dimensional photonic crystal is some analogy of the electronic band gap in a silicon crystal. Therefore, the photonic bandgap "controls" the light flux in a silicon photonic crystal in the same way as the transport of charge carriers occurs in a silicon crystal. In these two cases, the band gap is caused by standing waves of photons or electrons, respectively.

Make a photonic crystal yourself

Oddly enough, but the Maxwellian equations for photonic crystals are not sensitive to scaling, in contrast to the Schrödinger equation in the case of electronic crystals. This is due to the fact that the wavelength of an electron in a "normal" crystal is more or less fixed at a level of a few angstroms, while the dimensional scale of the wavelength of light in photonic crystals can vary from ultraviolet to microwave radiation, solely due to a change in the dimension of the photon components. gratings. This leads to truly inexhaustible possibilities for fine-tuning the properties of a photonic crystal.

At present, there are many methods for manufacturing photonic crystals. Some of them are more suitable for the formation of one-dimensional photonic crystals, others are convenient for two-dimensional ones, others are more applicable to three-dimensional photonic crystals, the fourth are used in the manufacture of photonic crystals on other optical devices, etc. However, not everything is limited only by varying the dimension of structural elements. Photonic crystals can also be created due to optical non-linearity, metal-non-metal transition, liquid crystal state, ferroelectric birefringence, swelling and shrinking of polymer gels, and so on, the main thing is to change the refractive index.

Where without defects?!

There are practically no materials in the world in which there would be no defects, and this is good. It is precisely the defects in solid-phase materials in b about to a greater extent than the crystal structure itself, influence the various properties of materials and, ultimately, their functional characteristics, as well as possible applications. A similar statement is also true in the case of photonic crystals. From a theoretical consideration, it follows that the introduction of defects (point, extended - dislocations - or bending) at the microlevel into an ideal photonic lattice allows you to create certain states inside the photonic bandgap, on which light can be localized, and the propagation of light can be limited or, on the contrary, enhanced along and around a very small waveguide (Figure 6). If we draw an analogy with semiconductors, then these states resemble impurity levels in semiconductors. Photonic crystals with such a “controlled imperfection” can be used to create all-optical devices and circuits of a new generation of optical telecommunication technologies.

Lighting informatics

Figure 7 shows one of the futuristic images of the all-light chip of the future, which has undoubtedly excited the imagination of chemists, physicists and materials scientists for a decade. The all-optical chip consists of integrated micro-sized photonic crystals with 1D, 2D and 3D periodicity, which can play the role of switches, filters, low-threshold lasers, etc., while light is transmitted between them through waveguides solely due to the defective structure. And although the topic of photonic crystals exists in the "roadmaps" for the development of photonic technologies, research and practical application of these materials are still at the very early stages of their development. This is the subject of future discoveries that may lead to the creation of all-light ultrafast computers, as well as quantum computers. However, in order for the dreams of science fiction writers and many scientists who have devoted their lives to studying such interesting and practically significant materials as photonic crystals to come true, a number of questions must be answered. For example, such as: what needs to be changed in the materials themselves in order to solve the problem associated with the reduction of such integrated chips from micro-sized photonic crystals for wide application in practice? Is it possible to use microdesign (“top-down”), or self-assembly (“bottom-up”), or some fusion of these two methods (for example, directed self-assembly) to commercialize the production of chips from microsized photonic crystals? Is the science of computers based on light chips made of microphotonic crystals a reality, or is it still a futurist fantasy?

The idea of ​​photonics of nanosized structures and photonic crystals was born while analyzing the possibility of creating an optical band structure. It was assumed that in the optical band structure, as well as in the semiconductor band structure, allowed and forbidden states for photons with different energies should exist. Theoretically, a model of the medium was proposed, in which periodic changes in the permittivity or refractive index of the medium were used as the periodic potential of the lattice. Thus, the concept of "photonic band gap" in a "photonic crystal" was introduced.

Photonic Crystal is a superlattice in which a field is artificially created, and its period is orders of magnitude greater than the period of the main lattice. A photonic crystal is a semitransparent dielectric with a certain periodic structure and unique optical properties.

The periodic structure is formed from the smallest holes, which periodically change the dielectric constant r. The diameter of these holes is such that light waves of a strictly defined length pass through them. All other waves are absorbed or reflected.

Photonic bands are formed in which the phase velocity of light propagation depends on e. In a crystal, light propagates coherently and forbidden frequencies appear, depending on the direction of propagation. Bragg diffraction for photonic crystals takes place in the optical wavelength range.

Such crystals are called photonic bandgap materials (PBGs). From the point of view of quantum electronics, Einstein's law for stimulated emission does not hold in such active media. In accordance with this law, the rates of induced emission and absorption are equal and the sum of the excited N 2 and unexcited

atoms JV is A, + N., = N. Then or 50%.

In photonic crystals, a 100% level population inversion is possible. This makes it possible to reduce the pump power and reduce the unnecessary heating of the crystal.

If the crystal is affected by sound waves, then the length of the light wave and the direction of movement of the light wave, characteristic of the crystal, can change. A distinctive property of photonic crystals is the proportionality of the reflection coefficient R light in the long-wavelength part of the spectrum to its frequency squared co 2, and not as for Rayleigh scattering R~ from 4 . The short-wave component of the optical spectrum is described by the laws of geometric optics.

In the industrial creation of photonic crystals, it is necessary to find a technology for creating three-dimensional superlattices. This is a very difficult task, since standard replication techniques using lithography methods are unacceptable for creating 3D nanostructures.

The attention of researchers was attracted by noble opal (Fig. 2.23). Is it a mineral Si() 2 ? P 1.0 hydroxide subclass. In natural opals, the voids of the globules are filled with silica and molecular water. From the point of view of nanoelectronics, opals are close-packed (mainly according to the cubic law) nanospheres (globules) of silica. As a rule, the diameter of nanospheres is in the range of 200–600 nm. The packing of silica globules forms a three-dimensional lattice. Such superlattices contain structural voids 140–400 nm in size, which can be filled with semiconductor, optically active, and magnetic materials. In an opal-like structure, it is possible to create a three-dimensional lattice with a nanoscale structure. The optical opal matrix structure can serve as a 3E photonic crystal.

The technology of oxidized macroporous silicon has been developed. Based on this technological process, three-dimensional structures in the form of silicon dioxide pins were created (Fig. 2.24).

Photonic band gaps were found in these structures. The band gap parameters can be changed at the stage of lithographic processes or by filling the pin structure with other materials.

Various designs of lasers have been developed on the basis of photonic crystals. Another class of optical elements based on photonic crystals is photonic crystal fibers(FKV). They have

Rice. 2.23. Structure of synthetic opal (a) and natural opals (b)"

" A source: Gudilin E. A.[and etc.]. Wealth of the Nanoworld. Photo essay from the depths of matter; ed. Yu. D. Tretyakova. M.: BINOM. Knowledge Lab, 2010.

Rice. 2.24.

band gap in a given wavelength range. Unlike conventional optical fibers, photonic bandgap fibers have the ability to shift the zero dispersion wavelength to the visible region of the spectrum. In this case, the conditions for soliton regimes of visible light propagation are provided.

By changing the size of the air tubes and, accordingly, the size of the core, it is possible to increase the concentration of the power of light radiation, the nonlinear properties of the fibers. By varying the fiber and cladding geometry, an optimal combination of strong non-linearity and low dispersion can be obtained in the desired wavelength range.

On fig. 2.25 are presented to the FCF. They are divided into two types. The first type is referred to FKV with a continuous light-guiding core. Structurally, such a fiber is made in the form of a core of quartz glass in a shell of a photonic crystal. The wave properties of such fibers are provided both by the effect of total internal reflection and by the band properties of the photonic crystal. Therefore, low-order modes propagate in such fibers in a wide spectral range. High-order modes are shifted into the shell and decay there. In this case, the waveguiding properties of the crystal for zero-order modes are determined by the effect of total internal reflection. The band structure of a photonic crystal manifests itself only indirectly.

The second type of FKV has a hollow light-guiding core. Light can propagate both through the core of the fiber and through the cladding. At the core of

Rice. 2.25.

a - section with a continuous light-guiding core;

6 - section with a hollow light-guiding residential strand, the refractive index is less than the average refractive index of the shell. This makes it possible to significantly increase the power of the transported radiation. At present, fibers have been created that have a loss of 0.58 dB / km at a wavelength X= 1.55 µm, which is close to the loss in standard single-mode fiber (0.2 dB/km).

Among other advantages of photonic crystal fibers, we note the following:

  • single-mode mode for all calculated wavelengths;
  • wide range of main fashion spot change;
  • constant and high value of the dispersion coefficient for wavelengths of 1.3-1.5 μm and zero dispersion for wavelengths in the visible spectrum;
  • controlled polarization values, group velocity dispersions, transmission spectrum.

Fibers with a photonic crystal cladding are widely used to solve problems in optics, laser physics, and especially in telecommunications systems. Recently, interest has been attracted by various resonances arising in photonic crystals. Polariton effects in photonic crystals take place during the interaction of electron and photon resonances. When creating metal-dielectric nanostructures with a period much smaller than the optical wavelength, it is possible to realize a situation in which the conditions r

A very significant product of the development of photonics are telecommunication fiber-optic systems. Their functioning is based on the processes of electro-optical conversion of an information signal, transmission of a modulated optical signal to a fiber optic light guide, and inverse opto-electronic conversion.